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Proportional Reasoning

Objective

Now let’s turn our attention to ratios and proportions. We’ll also discuss the idea of scale, and see how this applies to a geometric figure.

Previously Covered:

In the previous section, we refreshed your memory on bases, powers, exponents, roots, and their attendant operations.
We also covered the basics of scientific notation.

Ratios and Proportions

A ratio is a quotient used to compare two numbers.

A ratio can be represented:

As a fraction ()
With a colon (4 : 5)
With the word to (4 to 5)

Although a ratio may look like a fraction, it can have a larger first number (top) than second number (bottom).

A proportion is two or more ratios that are equivalent to each other. In a proportion, the cross products are equal. Cross products are the result of multiplying on the diagonal across the equal sign.

Since cross products must be equal in a proportion, you can use this property to solve for a missing piece of information in a proportion.

Step 1. Cross multiply.

Step 2. Use division to solve for the variable.

Let’s look at another proportion:

In this example, you can make equivalent fractions by determining what 5 is multiplied by to get 30. Because it is 6, you would also have to multiply 8 by 6, obtaining 48. Although this works in this example, when the answers are not integers, using cross multiplication is more efficient and accurate.

Solving More Complex Proportions

The method for solving proportions is always the same: cross multiply and isolate the variable. But it may take more than two steps to get the variable alone, as in the following examples.

Solve Example

Cross multiply: 20(x+4) = 90

Distributive Property: 20x + 80 = 90

Subtract 80 from both sides: 20x = 10

Divide both sides by 20: Solved

Question

The ratio of women to men at a party is 4 : 3. If there are 120 men at the party, how many women are there?

Reveal Answer

The correct answer is C. One way to set up the proportion is . Cross multiply to get 3w =480, and then divide both sides by three to get 160 women.

Blueprints and Miniatures: Utilizing Scale

Scale drawings, including maps, are common examples that require proportional reasoning. Sometimes you don’t have to go through all the formal steps of solving a proportion to find out the information you are seeking, but it’s proportional reasoning nonetheless.

Example

The scale on a map is 1 inch : 300 miles. If the distance from Phoenix to Los Angeles on this map is 2 1/4 inches, how many miles is it?

1 inch : 300 mi = 2.25 inches : x miles

x = 675 miles

The same process works for comparing the size of regular objects to models of those objects. Model trains, planes, and cars often state the ratio of model size to real size.

Example

A model car is five inches long and three inches wide. If the actual car is 17 feet long, how wide is it?

5w = 51

w = 10.2 feet

Question

A set of construction drawings for a house has a scale of 1 inch : 2 feet. The house is going to be 28 feet tall at its highest point. How many inches is that on the drawing?

14 inches
16 inches
28 inches
56 inches

Reveal Answer

The correct answer is A. The number of inches on the drawing is half the number of feet in reality, so divide 28 by 2.

Similar Triangles

With similar triangles and other similar figures, the lengths of the sides in the figures are in proportion to each other. There are a number of ways to set up the proportion for these types of problems, and the key is to keep like information together.

These are also similar triangles:

Let’s see how these work in practice. We’ll start with an old favorite.

A 25-foot tall flagpole is casting a shadow 30 feet long. A person standing nearby casts a shadow 7 feet long. How tall is the person?

Start by making a sketch:

Next, choose how to set up the proportion and solve it.

The person is about 5.8 feet tall.

Next, try this one yourself.

Question

Find the missing length on the pair of similar triangles below.

1.0 cm
2.7 cm
6.0 cm
13.5 cm

Reveal Answer

The correct choice is B. Set up the proportion and cross-multiply.
Solving the proportion

Review

A ratio is a quotient used to compare two numbers.
A proportion is two or more ratios that are equivalent to each other.
Solve simple proportions by cross multiplying to get an equation that can be solved by division.
Solve complex proportions in essentially the same way; it may take several steps to solve the equation that results from the cross-multiplication.
To solve problems about scale models and drawings, set up a proportion that you can solve. Keep corresponding information together when you set up the proportion.
Similar triangles provide another opportunity to use proportions to solve for the length of the missing side.

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